Decidability of the existential theory of the rationals (∃ℚ)

Determine whether the existential theory of the rational numbers is decidable.

Background

The authors discuss a potential connection between triangular grid embeddings of penny graphs and the existential theory of the rationals (∃ℚ). They note that if such a reduction existed, the resulting problem would be complete for ∃ℚ, whose decidability is a well-known open question in logic.